Given the following Bayessian Network:
I wonder when is it reasonable to estimate $p(u\mid c)$ as
$$ p(u\mid c) \approx p(c\mid w=w_1,\ldots,w_t)$$
I want to estimate that because I can't calculate $p(u\mid c)$ because $u$ is not observable. I've been looking for inference and reasoning in bayes networks but I couldn't find any inference like this.
I'm going to take a Bayesian stab at this:
By Bayes' rule $$ P(u \mid c) = \frac{P(c \mid u) P(u)}{P(c)} $$ $ P(c) $ is just a normalizing constant so what you're really interested in is $$ P(c \mid u) P(u) $$ The first term $ P(c \mid u) $ is the likelihood. This quantity depends on your model of how $ c $ is generated from $ u $. The second term, $ P(u) $ is your prior. This is what you believe the value of $ u $ before observing data.
If you can observe $ c $, then notice that this expression does not depend at all on $ w $ or $ q $. In other words, $ u \perp w,q \mid c $ ($ u $ is conditionally independent of $ w $ and $ q $ given $ c $)